Finite Group Databasebeta

← C3×C4×Q8×AGL(1,𝔽5) #14319619

Subgroups — C3×C4×Q8×AGL(1,𝔽5) #14319619

Order 1920.

1368 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C212yesmaximal
2C3×C4×Q8×D5C211yesmaximal, characteristic
3C3×C4⋊C4×AGL(1,𝔽5)C216yesmaximal
4C3×C4×C4⋊AGL(1,𝔽5)C213yesmaximal
5C3×C42×AGL(1,𝔽5)C213yesmaximal
6C4×Q8×AGL(1,𝔽5)C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C3×Q8×AGL(1,𝔽5)C4116yesdirect product factor
9C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
10C2×C3×C4×AGL(1,𝔽5)C2213yes—
11C2×C3×C4×AGL(1,𝔽5)C2216yes—
12C2×C3×C4×AGL(1,𝔽5)C2216yes—
13C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
14C3×C4×C5×Q8C411yescharacteristic, complemented
15C3×Q8×Dic5C411yescharacteristic, complemented
16C3×D5×C4⋊C4C2213yes—
17C3×#4033C413yescomplemented
18C3×C42×D5C2213yes—
19C3×C4×Dic10C413yescomplemented
20C3×(C2×AGL(1,𝔽5))⋊C4C2216yes—
21C3×C42×Q8—51nomaximal, Hall
22C2×Q8×AGL(1,𝔽5)C2×C312yes—
23C4×Q8×D5C2×C311yescharacteristic
24C4⋊C4×AGL(1,𝔽5)C2×C316yes—
25C4×C4⋊AGL(1,𝔽5)C2×C313yes—
26C42×AGL(1,𝔽5)C2×C313yes—
27C22×C3×AGL(1,𝔽5)C2316yes—
28C22×C3×AGL(1,𝔽5)C2312yes—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×Q8×D5C2×C418yes—
31C2×C3×C4×D5C2313yes—
32C2×C3×C4×D5C2313yes—
33C2×C3×C4×D5C2311yescharacteristic
34C2×C3×Dic10C2×C413yes—
35C3×C4⋊AGL(1,𝔽5)C2×C4124yes—
36C3×C4×AGL(1,𝔽5)—212no—
37C3×C4×AGL(1,𝔽5)—26no—
38C3×C4×AGL(1,𝔽5)C2×C4124yes—
39C3×C4×AGL(1,𝔽5)Q814yesdirect product factor
40C3×C5×C4⋊C4C2×C413yes—
41C3×C42×C5C2×C413yes—
42C3×C4⋊Dic5C2×C413yes—
43C3×#683C2×C416yes—
44C3×C4×Dic5C2×C413yes—
45C3×C4×Dic5C2×C413yes—
46C3×C4×Dic5C2×C413yes—
47C2×C3×C4×Q8—52no—
48C2×C3×C4×Q8—51no—
49C3×C4×C4⋊C4—53no—
50C3×C4×C4⋊C4—56no—
51C3×C43—53no—
52C2×Q8×D5C22×C311yescharacteristic
53Q8×AGL(1,𝔽5)C3×C4116yesdirect product factor
54C2×C4⋊AGL(1,𝔽5)C22×C316yes—
55C2×C4×AGL(1,𝔽5)C22×C313yes—
56C2×C4×AGL(1,𝔽5)C22×C316yes—
57C2×C4×AGL(1,𝔽5)C22×C316yes—
58C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
59C4×C5×Q8C3×C411yescharacteristic, complemented
60Q8×Dic5C3×C411yescharacteristic, complemented
61D5×C4⋊C4C22×C313yes—
62#4033C3×C413yescomplemented
63C42×D5C22×C313yes—
64C4×Dic10C3×C413yescomplemented
65(C2×AGL(1,𝔽5))⋊C4C22×C316yes—
66C42×Q8—51noSylow
67C22×C3×D5C2411yescharacteristic
68C2×C3×AGL(1,𝔽5)—26no—
69C2×C3×AGL(1,𝔽5)—26no—
70C2×C3×AGL(1,𝔽5)C22×C4112yes—
71C2×C3×AGL(1,𝔽5)C22×C414yes—
72C2×C3×AGL(1,𝔽5)C2×Q814yes—
73C2×C3×AGL(1,𝔽5)#4114yes—
74C3×C5×Q8C4214yescomplemented
75C2×C3×C4×C5C22×C413yes—
76C2×C3×C4×C5C22×C413yes—
77C2×C3×C4×C5C22×C411yescharacteristic
78C2×C3×Dic5C22×C413yes—
79C2×C3×Dic5C22×C413yes—
80C2×C3×Dic5C22×C411yescharacteristic
81C3×C4×D5—26no—
82C3×C4×D5C22×C4112yes—
83C3×C4×D5C2×Q812yes—
84C3×C4×D5#4112yes—
85C3×Dic10C42112yescomplemented
86C22×C3×Q8—51no—
87C3×C4×Q8—516no—
88C3×C4×Q8—51no—
89C3×C4×Q8—53no—
90C3×C4×Q8—53no—
91C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, direct product factor
92C2×C3×C4⋊C4—56no—
93C2×C3×C4⋊C4—53no—
94C2×C3×C42—51no—
95C2×C3×C42—56no—
96C2×C3×C42—53no—
97C2×C3×C42—56no—
98C2×C3×C42—53no—
99C3×#95—56no—
100C22×AGL(1,𝔽5)C23×C316yes—
101C22×AGL(1,𝔽5)C23×C312yes—
102C2×C5×Q8C2×C3×C411yescharacteristic
103Q8×D5C2×C3×C418yes—
104C2×C4×D5C23×C313yes—
105C2×C4×D5C23×C313yes—
106C2×C4×D5C23×C311yescharacteristic
107C2×Dic10C2×C3×C413yes—
108C4⋊AGL(1,𝔽5)C2×C3×C4124yes—
109C4×AGL(1,𝔽5)—212no—
110C4×AGL(1,𝔽5)—26no—
111C4×AGL(1,𝔽5)C2×C3×C4124yes—
112C4×AGL(1,𝔽5)C3×Q814yesdirect product factor
113C5×C4⋊C4C2×C3×C413yes—
114C42×C5C2×C3×C413yes—
115C4⋊Dic5C2×C3×C413yes—
116#683C2×C3×C416yes—
117C4×Dic5C2×C3×C413yes—
118C4×Dic5C2×C3×C413yes—
119C4×Dic5C2×C3×C413yes—
120C2×C4×Q8—52no—
121C2×C4×Q8—51no—
122C4×C4⋊C4—53no—
123C4×C4⋊C4—56no—
124C43—53no—
125C22×C3×C5C23×C411yescharacteristic
126C2×C3×D5C23×C412yes—
127C2×C3×D5C22×Q811yescharacteristic
128C2×C3×D5C2×#4112yes—
129C2×C3×D5C2×#4111yescharacteristic
130C3×AGL(1,𝔽5)—212no—
131C3×AGL(1,𝔽5)C4×Q818yesdirect product factor
132C3×C4×C5—23no—
133C3×C4×C5C2×C4216yes—
134C3×C4×C5C4×Q812yescomplemented
135C3×Dic5—23no—
136C3×Dic5C2×C4216yes—
137C3×Dic5C4×Q812yescomplemented
138C2×C3×Q8—58no—
139C2×C3×Q8—53no—
140C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
141C22×C3×C4—52no—
142C22×C3×C4—53no—
143C22×C3×C4—51no—
144C22×C3×C4—56no—
145C22×C3×C4—53no—
146C3×C4⋊C4—524no—
147C3×C4⋊C4—56no—
148C3×C4⋊C4—53no—
149C3×C4⋊C4C2×AGL(1,𝔽5)13yes—
150C3×C42—524no—
151C3×C42—54no—
152C3×C42—53no—
153C3×C42—53no—
154C3×C42—53no—
155C3×C42—1012no—
156C3×C42—106no—
157C3×C42C2×AGL(1,𝔽5)13yes—
158C22×D5C24×C311yescharacteristic
159C2×AGL(1,𝔽5)—26no—
160C2×AGL(1,𝔽5)—26no—
161C2×AGL(1,𝔽5)C22×C3×C4112yes—
162C2×AGL(1,𝔽5)C22×C3×C414yes—
163C2×AGL(1,𝔽5)C2×C3×Q814yes—
164C2×AGL(1,𝔽5)C3×#4114yes—
165C5×Q8C3×C4214yescomplemented
166C2×C4×C5C22×C3×C413yes—
167C2×C4×C5C22×C3×C413yes—
168C2×C4×C5C22×C3×C411yescharacteristic
169C2×Dic5C22×C3×C413yes—
170C2×Dic5C22×C3×C413yes—
171C2×Dic5C22×C3×C411yescharacteristic
172C4×D5—26no—
173C4×D5C22×C3×C4112yes—
174C4×D5C2×C3×Q812yes—
175C4×D5C3×#4112yes—
176Dic10C3×C42112yescomplemented
177C22×Q8—51no—
178C4×Q8—516no—
179C4×Q8—51no—
180C4×Q8—53no—
181C4×Q8—53no—
182C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, direct product factor
183C2×C4⋊C4—56no—
184C2×C4⋊C4—53no—
185C2×C42—56no—
186C2×C42—51no—
187C2×C42—53no—
188C2×C42—56no—
189C2×C42—53no—
190#95—56no—
191C2×C3×C5C22×C4211yescharacteristic
192C2×C3×C5C2×C4×Q811yescharacteristic
193C2×C3×C5C4×#4111yescharacteristic
194C3×D5C2×C4×Q812yes—
195C3×D5C4×#4112yes—
196C23×C3—51no—
197C3×Q8—512no—
198C3×Q8C4×AGL(1,𝔽5)14yesdirect product factor
199C2×C3×C4—54no—
200C2×C3×C4—54no—
201C2×C3×C4—54no—
202C2×C3×C4—51no—
203C2×C3×C4—53no—
204C2×C3×C4—512no—
205C2×C3×C4—52no—
206C2×C3×C4—52no—
207C2×C3×C4—512no—
208C2×C3×C4—53no—
209C2×C3×C4—106no—
210C2×C3×C4—106no—
211C2×C3×C4—106no—
212C2×C3×C4C22×AGL(1,𝔽5)13yes—
213C2×C3×C4C22×AGL(1,𝔽5)13yes—
214C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
215C22×C5C23×C3×C411yescharacteristic
216C2×D5C23×C3×C412yes—
217C2×D5C22×C3×Q811yescharacteristic
218C2×D5C2×C3×#4112yes—
219C2×D5C2×C3×#4111yescharacteristic
220AGL(1,𝔽5)—212no—
221AGL(1,𝔽5)C3×C4×Q818yesdirect product factor
222C4×C5—23no—
223C4×C5C2×C3×C4216yes—
224C4×C5C3×C4×Q812yescomplemented
225Dic5—23no—
226Dic5C2×C3×C4216yes—
227Dic5C3×C4×Q812yescomplemented
228C2×Q8—58no—
229C2×Q8—53no—
230C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
231C22×C4—52no—
232C22×C4—51no—
233C22×C4—53no—
234C22×C4—56no—
235C22×C4—53no—
236C4⋊C4—524no—
237C4⋊C4—56no—
238C4⋊C4—53no—
239C4⋊C4C2×C3×AGL(1,𝔽5)13yes—
240C42—524no—
241C42—54no—
242C42—53no—
243C42—53no—
244C42—53no—
245C42—1012no—
246C42—106no—
247C42C2×C3×AGL(1,𝔽5)13yes—
248C3×C5C42×Q811yescharacteristic, complemented, Hall
249C22×C3—52no—
250C22×C3—51no—
251C22×C3—51no—
252C22×C3—52no—
253C22×C3C23×AGL(1,𝔽5)11yescharacteristic
254C3×C4—23no—
255C3×C4—58no—
256C3×C4—56no—
257C3×C4—52no—
258C3×C4—1012no—
259C3×C4—103no—
260C3×C4C2×C4×AGL(1,𝔽5)16yes—
261C3×C4Q8×AGL(1,𝔽5)12yesdirect product factor
262C2×C5C22×C3×C4211yescharacteristic
263C2×C5C2×C3×C4×Q811yescharacteristic
264C2×C5C3×C4×#4111yescharacteristic
265D5C2×C3×C4×Q812yes—
266D5C3×C4×#4112yes—
267C23—51no—
268Q8—512no—
269Q8C3×C4×AGL(1,𝔽5)14yesdirect product factor
270C2×C4—54no—
271C2×C4—54no—
272C2×C4—54no—
273C2×C4—51no—
274C2×C4—53no—
275C2×C4—512no—
276C2×C4—52no—
277C2×C4—52no—
278C2×C4—512no—
279C2×C4—53no—
280C2×C4—106no—
281C2×C4—106no—
282C2×C4—106no—
283C2×C4C22×C3×AGL(1,𝔽5)13yes—
284C2×C4C22×C3×AGL(1,𝔽5)13yes—
285C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
286C2×C3—52no—
287C2×C3—52no—
288C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
289C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
290C2×C3#41×AGL(1,𝔽5)11yescharacteristic
291C5C3×C42×Q811yescharacteristic, complemented, Sylow
292C22—52no—
293C22—51no—
294C22—51no—
295C22—52no—
296C22C23×C3×AGL(1,𝔽5)11yescharacteristic
297C4—23no—
298C4—58no—
299C4—56no—
300C4—52no—
301C4—1012no—
302C4—103no—
303C4C2×C3×C4×AGL(1,𝔽5)16yes—
304C4C3×Q8×AGL(1,𝔽5)12yesdirect product factor
305C3C4×Q8×AGL(1,𝔽5)11yescharacteristic, direct product factor, Sylow
306C2—52no—
307C2—52no—
308C2C22×C3×C4×AGL(1,𝔽5)11yescharacteristic
309C2C2×C3×Q8×AGL(1,𝔽5)11yescharacteristic
310C2C3×#41×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3×C4C22×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C42
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C23×C4
prime radicalC22×C3×D5C24
cosocleC22×D5C24×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×C42×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C4×Q8×AGL(1,𝔽5)C3direct (N × H)
8. C3×Q8×AGL(1,𝔽5)C4direct (N × H)
14. C3×C4×C5×Q8C4semidirect
15. C3×Q8×Dic5C4semidirect
17. C3×#4033C4semidirect
19. C3×C4×Dic10C4semidirect
39. C3×C4×AGL(1,𝔽5)Q8direct (N × H)
53. Q8×AGL(1,𝔽5)C3×C4direct (N × H)
59. C4×C5×Q8C3×C4semidirect
60. Q8×Dic5C3×C4semidirect
62. #4033C3×C4semidirect
64. C4×Dic10C3×C4semidirect
74. C3×C5×Q8C42semidirect
85. C3×Dic10C42semidirect
91. C3×C4×Q8AGL(1,𝔽5)direct (N × H)
112. C4×AGL(1,𝔽5)C3×Q8direct (N × H)
131. C3×AGL(1,𝔽5)C4×Q8direct (N × H)
134. C3×C4×C5C4×Q8semidirect
137. C3×Dic5C4×Q8semidirect
165. C5×Q8C3×C42semidirect
176. Dic10C3×C42semidirect
182. C4×Q8C3×AGL(1,𝔽5)direct (N × H)
198. C3×Q8C4×AGL(1,𝔽5)direct (N × H)
221. AGL(1,𝔽5)C3×C4×Q8direct (N × H)
224. C4×C5C3×C4×Q8semidirect
227. Dic5C3×C4×Q8semidirect
248. C3×C5C42×Q8semidirect
261. C3×C4Q8×AGL(1,𝔽5)direct (N × H)
269. Q8C3×C4×AGL(1,𝔽5)direct (N × H)
291. C5C3×C42×Q8semidirect
304. C4C3×Q8×AGL(1,𝔽5)direct (N × H)
305. C3C4×Q8×AGL(1,𝔽5)direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C3×C4×Q8×D52
C3×C4⋊C4×AGL(1,𝔽5)2
C3×C4×C4⋊AGL(1,𝔽5)2
C3×C42×AGL(1,𝔽5)2
C4×Q8×AGL(1,𝔽5)3
C3×C42×Q85